{"id":63,"date":"2022-05-26T15:49:16","date_gmt":"2022-05-26T15:49:16","guid":{"rendered":"https:\/\/academic.csuohio.edu\/shao-sailai\/?page_id=63"},"modified":"2022-05-26T15:49:18","modified_gmt":"2022-05-26T15:49:18","slug":"mathematics-and-beauty","status":"publish","type":"page","link":"https:\/\/academic.csuohio.edu\/shao-sailai\/mathematics-and-beauty\/","title":{"rendered":"Mathematics and Beauty"},"content":{"rendered":"\n<h2 class=\"has-text-align-center wp-block-heading\">Mathematics &amp; Beauty<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"405\" height=\"405\" src=\"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-content\/uploads\/sites\/65\/2022\/05\/hopf.jpg\" alt=\"\" class=\"wp-image-64\" srcset=\"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-content\/uploads\/sites\/65\/2022\/05\/hopf.jpg 405w, https:\/\/academic.csuohio.edu\/shao-sailai\/wp-content\/uploads\/sites\/65\/2022\/05\/hopf-300x300.jpg 300w, https:\/\/academic.csuohio.edu\/shao-sailai\/wp-content\/uploads\/sites\/65\/2022\/05\/hopf-150x150.jpg 150w\" sizes=\"auto, (max-width: 405px) 100vw, 405px\" \/><\/figure><\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong><a href=\"https:\/\/academic.csuohio.edu\/shao_s\/KenShoemake.htm\">The figure (created by Ken Shoemake)<\/a>\u00a0is a visualization of the Hopf fibration of the three sphere over the two sphere. It shows both S^3 and S^2, and visually connect fibers to base points. For each point on S^2 there is a unique RGB color by putting the sphere inside a color cube. Each fiber is colored uniformly with the color of the base point. To display S^3, the author punctured and flattened it to B^3 using a unit quaternions for the S^3 and took their logarithms (the inverse of the exponential map at 1+0i+0j+0k) for B^3. To show structure, a few arcs of lattitude as base points were chosen to give sliced tori for fibers. The coloring is continuous on S^2, so it is also continuous and delightful in the fibers. It shows nesting tori, linking fibers, base-fiber relations.A visualization of the fibering of RP3 by S1 over S2 is explored in detail in the article by Rick Kreminski in Mathematics in Education and Research , vol. 6, no. 1 and his wonderful graphic which was on the cover of the Notices of the AMS in May 1997 can be found at the web site:<\/strong> <strong><a href=\"http:\/\/www.youtube.com\/watch?v=1yJ5EvV0K70\">Hopf fibration video <\/a><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"382\" src=\"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-content\/uploads\/sites\/65\/2022\/05\/viking1.gif\" alt=\"\" class=\"wp-image-65\"\/><\/figure><\/div>\n\n\n\n<p><strong>This Viking ship was created by Pontus Axelsson and Fredrik Sandberg (Using MATLAB) of\u00a0<a href=\"http:\/\/www.nada.kth.se\/na\/\">Department of Numerical Analysis and Computing Science <\/a>\u00a0of the Royal Institute of Technology in Stockholm, Sweden. This boat started its life as a rowboat for a project in a numerical methods course, but it grew until it was a fully-rigged Viking ship. There is also a movie (98K, mpg) of the boat in motion!<\/strong><\/p>\n\n\n<div class=\"wp-block-ub-divider ub_divider ub-divider-orientation-horizontal\" id=\"ub_divider_857d9f63-ebb9-4c47-8ff2-1994d63fea95\"><div class=\"ub_divider_wrapper\" style=\"position: relative; margin-bottom: 2px; width: 100%; height: 2px; \" data-divider-alignment=\"center\"><div class=\"ub_divider_line\" style=\"border-top: 2px solid #ccc; margin-top: 2px; \"><\/div><\/div><\/div>\n\n\n<p>More mathematical imaginary on the AMS Website (<a href=\"http:\/\/www.ams.org\/mathimagery\">&nbsp;www.ams.org\/mathimagery <\/a>), which mathematicians and artists create stunning works in all media and explore the visualization of mathematics.<br>video clips:&nbsp;<a href=\"http:\/\/www.youtube.com\/watch?v=V1RxTYRKMlY&amp;feature=related\">Math is Beautiful <\/a><br>How to Turn a Sphere Inside Out:&nbsp;<a href=\"http:\/\/www.youtube.com\/watch?v=R_w4HYXuo9M&amp;feature=related\">Part I <\/a><a href=\"http:\/\/www.youtube.com\/watch?v=x7d13SgqUXg&amp;feature=related\">Part II <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Mathematics &amp; Beauty The figure (created by Ken Shoemake)\u00a0is a visualization of the Hopf fibration of the three sphere over the two sphere. It shows both S^3 and S^2, and visually connect fibers to base points. For each point on S^2 there is a unique RGB color by putting the sphere inside a color cube.&mldr;<\/p>\n","protected":false},"author":10,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_relevanssi_hide_post":"","_relevanssi_hide_content":"","_relevanssi_pin_for_all":"","_relevanssi_pin_keywords":"","_relevanssi_unpin_keywords":"","_relevanssi_related_keywords":"","_relevanssi_related_include_ids":"","_relevanssi_related_exclude_ids":"","_relevanssi_related_no_append":"","_relevanssi_related_not_related":"","_relevanssi_related_posts":"","_relevanssi_noindex_reason":"","footnotes":""},"class_list":["post-63","page","type-page","status-publish","hentry"],"featured_image_src":null,"_links":{"self":[{"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/pages\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/comments?post=63"}],"version-history":[{"count":1,"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/pages\/63\/revisions"}],"predecessor-version":[{"id":66,"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/pages\/63\/revisions\/66"}],"wp:attachment":[{"href":"https:\/\/academic.csuohio.edu\/shao-sailai\/wp-json\/wp\/v2\/media?parent=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}